Relativity : On length contraction

[Delzac]

Homework Statement

The distance between the Earth and Alpha Centauri is 4.2 light years (1 light year is the distance traveled by light in one year). If astronauts could travel at v = 0.95c, then we on Earth would assume that the trip would take the astronauts 4.2 / 0.95 = 4.4 years. The astronauts however, disagree.
a) How much time passes on the astronauts clock?
b) What distance to Alpha Centauri do the astronauts measure?

Homework Equations

$$\Delta T = \gamma \Delta T_0$$
$$L = L_0/\gamma$$

The Attempt at a Solution

a)
$$v=0.95c \Delta T=4.4$$
Sub the values into equation you get $$\Delta T_0 = 1.37$$

b)

We have $$L_0=4.2 v=0.95c$$
Sub into equation we have L = 1.311 light years

Another Qns :
Exam smart, $$L < L_O$$ once again?
Also, is $$L_O = 4.2$$ because we are measuring the distance between the destination, and we are at rest relative to the 2 location?

 

[Hootenanny]

Careful, you have your times mixed up! What is the definition of proper time?

 

[Delzac]

Proper time is the time interval measured in the rest frame of an event(s)

But shouldn't the time that the astronauts experience be $$\Delta T_0$$?

 

[Delzac]

And didn't T > T_0 fit nicely?

 

[Hootenanny]

Proper time is the time interval measured in the rest frame of an event(s)

Correct!

But shouldn't the time that the astronauts experience be $$\Delta T_0$$?

What are the two events in question?

 

[Delzac]

Hmmm...so there are 2 events happening concurrently? never thought of that before

Event 1 : The astronauts moving to their destination?
Event 2 : No idea

 

[Hootenanny]

Okay, I'll help you out;

Event 1: The astronauts leave earth.
Event 2: The astronauts arrive at Alpha Centauri

Have you learned the lorentz transformations yet?

 

[Delzac]

Hmmm...not really, but at least i know the equation when i checked up wiki.
The lecture didn't cover lorentz transformations but from the looks of it, do we need to use lorentz transformation?(guess so)

 

[Hootenanny]

Oke doke, its just I feel the Lorentz transformations are a little more intuitive than the time dilation formula. So, now you've got your two events, can you tell me the proper time?

 

[Delzac]

From the 2 events you stated, I would still say proper time = time i measured?

This is so because since proper time = measured in the rest frame of an events.
Rest frame is me measuring the time.

So this Qns is different from the "Relativity : SpaceCraft Qns" thread because what happen for "Relativity : SpaceCraft Qns" is internal(loosely speaking) events, since it happens inside the craft.

But again, if my ans is wrong why is it that this "rule", T > T_0, is seemingly not contradicted?

Usually when i sub wrong values, T > T_0 is not obeyed.

 

[Delzac]

And BTW is using lorentz transformation better then using time dilation formula?

 

[Delzac]

Hmmmm come to think of it why can't the event be :

Event : the craft moved for time $$\Delta T$$?

If this is the case, then rest frame of proper time will be astronaut's times.

 

[Hootenanny]

An event in SR is a set of spacetime coordinates (x,y,z,t), in otherwords, something happened at (x,y,z) at time t.

 

[Delzac]

So by that you are defining that event is just an instant? and not a duration like i have mention "Event : the craft moved for time T"

Therefore the above event is not an event, thus proper time isn't astronaut's time?

 

[Hootenanny]

Yes, think of an event as a point in space and time. Therefore, we must consider two events which I outlined above.

 

[Delzac]

therefore proper time is = 4.4

using time dilation formula, i will obtain $$\Delta T=14.09years$$ which is what the astronaut experience.

 

[Delzac]

And i think part (b) of my answer also shouldn't have a problem?

 

[Hootenanny]

My apologies, I've got this question totally backwards. You had the correct answer in your original post. My bad. I'm gona go get some sleep now :zzz: I'm really sorry to have wasted your time

 

[Delzac]

Nah its ok, when you are back can you explain what you meant by " got the question totally backwards".

How should the question be rephrase so that your working/steps are correct?

 

[Delzac]

And also explain why your argument/steps don't apply for the original question?

 

[Hootenanny]

I've got it totally backwards because I messed up the reference frames. The two events (departing Earth and arriving at Alpha Centauri) both occurred at the the same location in the astronauts reference frame, therefore this is the proper time. Hence, the time observed on Earth is the dilated time. Thus the proper time, experienced by the astronauts was;

$$\Delta t_{0} = \frac{\Delta t}{\gamma} = 1.37$$

As you correctly say. My bad.